Properties

Label 6.6.11796113.1
Degree $6$
Signature $[6, 0]$
Discriminant $7^{4}\cdot 17^{3}$
Root discriminant $15.09$
Ramified primes $7, 17$
Class number $1$
Class group Trivial
Galois Group $C_6$ (as 6T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -25, 57, 11, -17, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 17*x^4 + 11*x^3 + 57*x^2 - 25*x - 13)
gp: K = bnfinit(x^6 - x^5 - 17*x^4 + 11*x^3 + 57*x^2 - 25*x - 13, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut 17 x^{4} \) \(\mathstrut +\mathstrut 11 x^{3} \) \(\mathstrut +\mathstrut 57 x^{2} \) \(\mathstrut -\mathstrut 25 x \) \(\mathstrut -\mathstrut 13 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[6, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(11796113=7^{4}\cdot 17^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $15.09$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 17$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(16,·)$, $\chi_{119}(1,·)$, $\chi_{119}(50,·)$, $\chi_{119}(67,·)$, $\chi_{119}(86,·)$, $\chi_{119}(18,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1651} a^{5} + \frac{93}{1651} a^{4} + \frac{470}{1651} a^{3} - \frac{386}{1651} a^{2} + \frac{95}{1651} a + \frac{50}{127}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $5$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{54}{1651} a^{5} + \frac{69}{1651} a^{4} - \frac{1036}{1651} a^{3} - \frac{1032}{1651} a^{2} + \frac{3479}{1651} a + \frac{33}{127} \),  \( \frac{34}{1651} a^{5} - \frac{140}{1651} a^{4} - \frac{530}{1651} a^{3} + \frac{1735}{1651} a^{2} + \frac{1579}{1651} a - \frac{205}{127} \),  \( \frac{103}{1651} a^{5} - \frac{327}{1651} a^{4} - \frac{1120}{1651} a^{3} + \frac{3168}{1651} a^{2} - \frac{121}{1651} a - \frac{57}{127} \),  \( \frac{108}{1651} a^{5} + \frac{138}{1651} a^{4} - \frac{2072}{1651} a^{3} - \frac{2064}{1651} a^{2} + \frac{8609}{1651} a + \frac{193}{127} \),  \( \frac{54}{1651} a^{5} + \frac{69}{1651} a^{4} - \frac{1036}{1651} a^{3} - \frac{1032}{1651} a^{2} + \frac{1828}{1651} a + \frac{33}{127} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 33.1107945948 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra: \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{17}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7_17.6t1.1c1$1$ $ 7 \cdot 17 $ $x^{6} - x^{5} - 17 x^{4} + 11 x^{3} + 57 x^{2} - 25 x - 13$ $C_6$ (as 6T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7_17.6t1.1c2$1$ $ 7 \cdot 17 $ $x^{6} - x^{5} - 17 x^{4} + 11 x^{3} + 57 x^{2} - 25 x - 13$ $C_6$ (as 6T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.